Diffusion-driven instability of topological signals coupled by the Dirac operator.

Area of Science:

• Complex systems
• Network theory
• Mathematical biology

Background:

• Reaction-diffusion systems are key to understanding nonlinear processes in discrete systems like the brain.
• Traditionally, these systems were studied only on network nodes.
• Real-world systems involve dynamical variables on nodes, links, and higher-dimensional cells, creating topological signals.

Purpose of the Study:

• To investigate reaction-diffusion processes of topological signals coupled through the Dirac operator.
• To establish conditions for Turing pattern emergence in these systems.
• To analyze the localization and projection of Turing patterns on network components.

Main Methods:

• Utilizing the Dirac operator to couple topological signals of different dimensions.
• Focusing on networks comprising nodes and links.
• Analyzing pattern formation and projection properties.

Main Results:

• Turing patterns emerge from reaction-diffusion processes on topological signals.
• These patterns are never localized solely on nodes or links.
• Projections of topological signals displaying Turing patterns also exhibit these patterns.

Conclusions:

• The Dirac operator enables cross-diffusion between topological signals of varying dimensions.
• The study provides a theoretical framework and validation for Turing patterns in topological reaction-diffusion systems.
• Findings offer new insights into nonlinear dynamics in complex network topologies.